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General Orthogonal Polynomials
- in “Encyclopedia of Mathematics and its Applications,” 43
, 1992
"... Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. ..."
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Cited by 92 (8 self)
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Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
Double scaling limit in the random matrix model: the Riemann-Hilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 81 (10 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1, 1]
"... We consider polynomials that are orthogonal on [-1, 1] with respect to a modified Jacobi weight (1 - x) # (1 + x) # h(x), with #, # > -1 and h real analytic and stricly positive on [-1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval ..."
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Cited by 73 (24 self)
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We consider polynomials that are orthogonal on [-1, 1] with respect to a modified Jacobi weight (1 - x) # (1 + x) # h(x), with #, # > -1 and h real analytic and stricly positive on [-1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [-1, 1], for the recurrence coe#cients and for the leading coe#cients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for Riemann--Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szego function associated with the weight and for the local analysis around the endpoints 1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. 1 Supported by FWO research project G.0176.02 and by INTAS project 00-272 2 Supported by NSF grant #DMS-9970328 3 Supported by FWO research project G.0184.01 and by INTAS project 00-272 4 Research Assistant of the Fund for Scientific Research -- Flanders (Belgium) 1 1
Riemann-Hilbert problems for multiple orthogonal polynomials
, 2000
"... In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 2 matrix functions) associated which a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on ortho ..."
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Cited by 60 (26 self)
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In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 2 matrix functions) associated which a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on orthogonal polynomials, in particular strong asymptotics which hold uniformly in the complex plane. In this paper we will show that a similar Riemann-Hilbert problem (for (r + 1) (r + 1) matrix functions) is associated with multiple orthogonal polynomials. We show how this helps in understanding the relation between two types of multiple orthogonal polynomials and the higher order recurrence relations for these polynomials. Finally we indicate how an extremal problem for vector potentials is important for the normalization of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials. 1 Introduction Recently it was observed that ...
Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann–Hilbert problem
, 2002
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Random words, Toeplitz determinants and integrable systems
- I
, 2001
"... Abstract. It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the ..."
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Cited by 38 (7 self)
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Abstract. It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane. 1.
